Hubbard model diagonalization in 1D K-space for spinless Fermions

[Luqman Saleem]

I am trying to diagonalize hubbard model in real and K-space for spinless fermions. Hubbard model in real space is given as:
$$H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j)$$
I solved this Hamiltonian using MATLAB. It was quite simple. t and U are hopping and interaction potentials. c, $$c^\dagger$$ and n are annihilation, creation and number operators in real space respectively. The first term is hopping and 2nd is two-body interaction term. <i,j> is indicating that hopping is possible only to nearest neighbors. To solve this Hamiltonian I break it down as: (for M=# for sites=2 and N=# of particles=1)
$$H=-t (c_1^\dagger c_2 + c_2^\dagger c_1)+U n_1 n_2$$
The basis vectors that can be written in binary notation are:
01, 10
Using t=1, U=1 and above basis the Hamiltonian can be written as:
H=[0 -1
-1 0]
That is correct.
I checked with different values of M,N,U and t this MATLAB program give correct results.

$$**In K-space**$$
To diagonalize this Hamiltonian in K-space we can perform Fourier transform of operators that will results in:
$$H(k)=\sum_k \epsilon_k n_k + U / L \sum_ {k,k,q} c_k^\dagger c_{k-q} c_{k'}^\dagger c_{k'+q}$$
Where $$\epsilon_k=-2tcos(k)$$.
To diagonalize this Hamiltonian I make basis by taking k-points between -pi and +pi (first brillion zone) i.e. for M=2 and N=1 allowed k-points are: [0,pi]
Here first term is simple to solve and I have solved it already but I can't solve the 2nd term as it includes summation over three variables.
To get in more details of my attempt you can see https://physics.stackexchange.com/q/352833/140141

$$**My question:**$$
1. What is physical significance of 2nd term in H(k) given above? I mean what is it telling about which particles are hopping from where to where? What are limits on q, k and k'?
2. If you think any article can help me with this problem then please tell me about that.Thanks a lot.

 

[azntoon]

1. The second term in the Hamiltonian describes the scattering between two particles with momentum k and k' due to an interaction U. The limit on q is that it must be less than the first Brillouin zone (i.e. q must be between -pi and +pi). The limits on k and k' are determined by the wave vector of the two particles, i.e. they must both be within the allowed range of wave vectors.2. There are several articles which discuss the diagonalization of the Hubbard model in k-space. One example is “Diagonalization of the Hubbard Model in K Space” by A. G. Green, published in the Journal of Mathematical Physics in 1998.